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Vector-valued modular forms and Poincaré series. (English) Zbl 1145.11309

Summary: We initiate a general theory of vector-valued modular forms associated to a finite-dimensional representation \(\rho\) of \(\text{SL}(2, \mathbb Z)\). We introduce vector-valued Poincaré series and Eisenstein series and a version of the Petersson inner product, and establish analogs of basic results from the classical theory of modular forms concerning these objects, at least if the weight is large enough. In particular, we show that the space of entire vector-valued modular forms of weight \(k\) associated to \(\rho\) is a finite-dimensional vector space which, for large enough \(k\), is nonzero and spanned by Poincaré series. We show that Hecke’s estimate \(a_n=O(n^{k-1})\) continues to apply to the Fourier coefficients of component functions of entire vector-valued modular forms associated to \(\rho\) for large enough \(k\).

MSC:

11F11 Holomorphic modular forms of integral weight
11F99 Discontinuous groups and automorphic forms
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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